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## G = C3×C8⋊D7order 336 = 24·3·7

### Direct product of C3 and C8⋊D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C3×C8⋊D7
 Chief series C1 — C7 — C14 — C28 — C84 — C12×D7 — C3×C8⋊D7
 Lower central C7 — C14 — C3×C8⋊D7
 Upper central C1 — C12 — C24

Generators and relations for C3×C8⋊D7
G = < a,b,c,d | a3=b8=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Smallest permutation representation of C3×C8⋊D7
On 168 points
Generators in S168
(1 132 165)(2 133 166)(3 134 167)(4 135 168)(5 136 161)(6 129 162)(7 130 163)(8 131 164)(9 116 72)(10 117 65)(11 118 66)(12 119 67)(13 120 68)(14 113 69)(15 114 70)(16 115 71)(17 124 150)(18 125 151)(19 126 152)(20 127 145)(21 128 146)(22 121 147)(23 122 148)(24 123 149)(25 157 88)(26 158 81)(27 159 82)(28 160 83)(29 153 84)(30 154 85)(31 155 86)(32 156 87)(33 80 96)(34 73 89)(35 74 90)(36 75 91)(37 76 92)(38 77 93)(39 78 94)(40 79 95)(41 140 104)(42 141 97)(43 142 98)(44 143 99)(45 144 100)(46 137 101)(47 138 102)(48 139 103)(49 61 112)(50 62 105)(51 63 106)(52 64 107)(53 57 108)(54 58 109)(55 59 110)(56 60 111)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136)(137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152)(153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168)
(1 47 36 71 152 157 62)(2 48 37 72 145 158 63)(3 41 38 65 146 159 64)(4 42 39 66 147 160 57)(5 43 40 67 148 153 58)(6 44 33 68 149 154 59)(7 45 34 69 150 155 60)(8 46 35 70 151 156 61)(9 20 81 106 133 139 76)(10 21 82 107 134 140 77)(11 22 83 108 135 141 78)(12 23 84 109 136 142 79)(13 24 85 110 129 143 80)(14 17 86 111 130 144 73)(15 18 87 112 131 137 74)(16 19 88 105 132 138 75)(25 50 165 102 91 115 126)(26 51 166 103 92 116 127)(27 52 167 104 93 117 128)(28 53 168 97 94 118 121)(29 54 161 98 95 119 122)(30 55 162 99 96 120 123)(31 56 163 100 89 113 124)(32 49 164 101 90 114 125)
(1 62)(2 59)(3 64)(4 61)(5 58)(6 63)(7 60)(8 57)(9 13)(11 15)(17 73)(18 78)(19 75)(20 80)(21 77)(22 74)(23 79)(24 76)(25 102)(26 99)(27 104)(28 101)(29 98)(30 103)(31 100)(32 97)(33 145)(34 150)(35 147)(36 152)(37 149)(38 146)(39 151)(40 148)(41 159)(42 156)(43 153)(44 158)(45 155)(46 160)(47 157)(48 154)(49 168)(50 165)(51 162)(52 167)(53 164)(54 161)(55 166)(56 163)(66 70)(68 72)(81 143)(82 140)(83 137)(84 142)(85 139)(86 144)(87 141)(88 138)(89 124)(90 121)(91 126)(92 123)(93 128)(94 125)(95 122)(96 127)(105 132)(106 129)(107 134)(108 131)(109 136)(110 133)(111 130)(112 135)(114 118)(116 120)

G:=sub<Sym(168)| (1,132,165)(2,133,166)(3,134,167)(4,135,168)(5,136,161)(6,129,162)(7,130,163)(8,131,164)(9,116,72)(10,117,65)(11,118,66)(12,119,67)(13,120,68)(14,113,69)(15,114,70)(16,115,71)(17,124,150)(18,125,151)(19,126,152)(20,127,145)(21,128,146)(22,121,147)(23,122,148)(24,123,149)(25,157,88)(26,158,81)(27,159,82)(28,160,83)(29,153,84)(30,154,85)(31,155,86)(32,156,87)(33,80,96)(34,73,89)(35,74,90)(36,75,91)(37,76,92)(38,77,93)(39,78,94)(40,79,95)(41,140,104)(42,141,97)(43,142,98)(44,143,99)(45,144,100)(46,137,101)(47,138,102)(48,139,103)(49,61,112)(50,62,105)(51,63,106)(52,64,107)(53,57,108)(54,58,109)(55,59,110)(56,60,111), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168), (1,47,36,71,152,157,62)(2,48,37,72,145,158,63)(3,41,38,65,146,159,64)(4,42,39,66,147,160,57)(5,43,40,67,148,153,58)(6,44,33,68,149,154,59)(7,45,34,69,150,155,60)(8,46,35,70,151,156,61)(9,20,81,106,133,139,76)(10,21,82,107,134,140,77)(11,22,83,108,135,141,78)(12,23,84,109,136,142,79)(13,24,85,110,129,143,80)(14,17,86,111,130,144,73)(15,18,87,112,131,137,74)(16,19,88,105,132,138,75)(25,50,165,102,91,115,126)(26,51,166,103,92,116,127)(27,52,167,104,93,117,128)(28,53,168,97,94,118,121)(29,54,161,98,95,119,122)(30,55,162,99,96,120,123)(31,56,163,100,89,113,124)(32,49,164,101,90,114,125), (1,62)(2,59)(3,64)(4,61)(5,58)(6,63)(7,60)(8,57)(9,13)(11,15)(17,73)(18,78)(19,75)(20,80)(21,77)(22,74)(23,79)(24,76)(25,102)(26,99)(27,104)(28,101)(29,98)(30,103)(31,100)(32,97)(33,145)(34,150)(35,147)(36,152)(37,149)(38,146)(39,151)(40,148)(41,159)(42,156)(43,153)(44,158)(45,155)(46,160)(47,157)(48,154)(49,168)(50,165)(51,162)(52,167)(53,164)(54,161)(55,166)(56,163)(66,70)(68,72)(81,143)(82,140)(83,137)(84,142)(85,139)(86,144)(87,141)(88,138)(89,124)(90,121)(91,126)(92,123)(93,128)(94,125)(95,122)(96,127)(105,132)(106,129)(107,134)(108,131)(109,136)(110,133)(111,130)(112,135)(114,118)(116,120)>;

G:=Group( (1,132,165)(2,133,166)(3,134,167)(4,135,168)(5,136,161)(6,129,162)(7,130,163)(8,131,164)(9,116,72)(10,117,65)(11,118,66)(12,119,67)(13,120,68)(14,113,69)(15,114,70)(16,115,71)(17,124,150)(18,125,151)(19,126,152)(20,127,145)(21,128,146)(22,121,147)(23,122,148)(24,123,149)(25,157,88)(26,158,81)(27,159,82)(28,160,83)(29,153,84)(30,154,85)(31,155,86)(32,156,87)(33,80,96)(34,73,89)(35,74,90)(36,75,91)(37,76,92)(38,77,93)(39,78,94)(40,79,95)(41,140,104)(42,141,97)(43,142,98)(44,143,99)(45,144,100)(46,137,101)(47,138,102)(48,139,103)(49,61,112)(50,62,105)(51,63,106)(52,64,107)(53,57,108)(54,58,109)(55,59,110)(56,60,111), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168), (1,47,36,71,152,157,62)(2,48,37,72,145,158,63)(3,41,38,65,146,159,64)(4,42,39,66,147,160,57)(5,43,40,67,148,153,58)(6,44,33,68,149,154,59)(7,45,34,69,150,155,60)(8,46,35,70,151,156,61)(9,20,81,106,133,139,76)(10,21,82,107,134,140,77)(11,22,83,108,135,141,78)(12,23,84,109,136,142,79)(13,24,85,110,129,143,80)(14,17,86,111,130,144,73)(15,18,87,112,131,137,74)(16,19,88,105,132,138,75)(25,50,165,102,91,115,126)(26,51,166,103,92,116,127)(27,52,167,104,93,117,128)(28,53,168,97,94,118,121)(29,54,161,98,95,119,122)(30,55,162,99,96,120,123)(31,56,163,100,89,113,124)(32,49,164,101,90,114,125), (1,62)(2,59)(3,64)(4,61)(5,58)(6,63)(7,60)(8,57)(9,13)(11,15)(17,73)(18,78)(19,75)(20,80)(21,77)(22,74)(23,79)(24,76)(25,102)(26,99)(27,104)(28,101)(29,98)(30,103)(31,100)(32,97)(33,145)(34,150)(35,147)(36,152)(37,149)(38,146)(39,151)(40,148)(41,159)(42,156)(43,153)(44,158)(45,155)(46,160)(47,157)(48,154)(49,168)(50,165)(51,162)(52,167)(53,164)(54,161)(55,166)(56,163)(66,70)(68,72)(81,143)(82,140)(83,137)(84,142)(85,139)(86,144)(87,141)(88,138)(89,124)(90,121)(91,126)(92,123)(93,128)(94,125)(95,122)(96,127)(105,132)(106,129)(107,134)(108,131)(109,136)(110,133)(111,130)(112,135)(114,118)(116,120) );

G=PermutationGroup([(1,132,165),(2,133,166),(3,134,167),(4,135,168),(5,136,161),(6,129,162),(7,130,163),(8,131,164),(9,116,72),(10,117,65),(11,118,66),(12,119,67),(13,120,68),(14,113,69),(15,114,70),(16,115,71),(17,124,150),(18,125,151),(19,126,152),(20,127,145),(21,128,146),(22,121,147),(23,122,148),(24,123,149),(25,157,88),(26,158,81),(27,159,82),(28,160,83),(29,153,84),(30,154,85),(31,155,86),(32,156,87),(33,80,96),(34,73,89),(35,74,90),(36,75,91),(37,76,92),(38,77,93),(39,78,94),(40,79,95),(41,140,104),(42,141,97),(43,142,98),(44,143,99),(45,144,100),(46,137,101),(47,138,102),(48,139,103),(49,61,112),(50,62,105),(51,63,106),(52,64,107),(53,57,108),(54,58,109),(55,59,110),(56,60,111)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136),(137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152),(153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168)], [(1,47,36,71,152,157,62),(2,48,37,72,145,158,63),(3,41,38,65,146,159,64),(4,42,39,66,147,160,57),(5,43,40,67,148,153,58),(6,44,33,68,149,154,59),(7,45,34,69,150,155,60),(8,46,35,70,151,156,61),(9,20,81,106,133,139,76),(10,21,82,107,134,140,77),(11,22,83,108,135,141,78),(12,23,84,109,136,142,79),(13,24,85,110,129,143,80),(14,17,86,111,130,144,73),(15,18,87,112,131,137,74),(16,19,88,105,132,138,75),(25,50,165,102,91,115,126),(26,51,166,103,92,116,127),(27,52,167,104,93,117,128),(28,53,168,97,94,118,121),(29,54,161,98,95,119,122),(30,55,162,99,96,120,123),(31,56,163,100,89,113,124),(32,49,164,101,90,114,125)], [(1,62),(2,59),(3,64),(4,61),(5,58),(6,63),(7,60),(8,57),(9,13),(11,15),(17,73),(18,78),(19,75),(20,80),(21,77),(22,74),(23,79),(24,76),(25,102),(26,99),(27,104),(28,101),(29,98),(30,103),(31,100),(32,97),(33,145),(34,150),(35,147),(36,152),(37,149),(38,146),(39,151),(40,148),(41,159),(42,156),(43,153),(44,158),(45,155),(46,160),(47,157),(48,154),(49,168),(50,165),(51,162),(52,167),(53,164),(54,161),(55,166),(56,163),(66,70),(68,72),(81,143),(82,140),(83,137),(84,142),(85,139),(86,144),(87,141),(88,138),(89,124),(90,121),(91,126),(92,123),(93,128),(94,125),(95,122),(96,127),(105,132),(106,129),(107,134),(108,131),(109,136),(110,133),(111,130),(112,135),(114,118),(116,120)])

102 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 6A 6B 6C 6D 7A 7B 7C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 14A 14B 14C 21A ··· 21F 24A 24B 24C 24D 24E 24F 24G 24H 28A ··· 28F 42A ··· 42F 56A ··· 56L 84A ··· 84L 168A ··· 168X order 1 2 2 3 3 4 4 4 6 6 6 6 7 7 7 8 8 8 8 12 12 12 12 12 12 14 14 14 21 ··· 21 24 24 24 24 24 24 24 24 28 ··· 28 42 ··· 42 56 ··· 56 84 ··· 84 168 ··· 168 size 1 1 14 1 1 1 1 14 1 1 14 14 2 2 2 2 2 14 14 1 1 1 1 14 14 2 2 2 2 ··· 2 2 2 2 2 14 14 14 14 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 D7 M4(2) D14 C3×D7 C3×M4(2) C4×D7 C6×D7 C8⋊D7 C12×D7 C3×C8⋊D7 kernel C3×C8⋊D7 C3×C7⋊C8 C168 C12×D7 C8⋊D7 C3×Dic7 C6×D7 C7⋊C8 C56 C4×D7 Dic7 D14 C24 C21 C12 C8 C7 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 3 2 3 6 4 6 6 12 12 24

Matrix representation of C3×C8⋊D7 in GL2(𝔽13) generated by

 9 0 0 9
,
 11 10 4 2
,
 0 11 7 7
,
 6 8 7 7
G:=sub<GL(2,GF(13))| [9,0,0,9],[11,4,10,2],[0,7,11,7],[6,7,8,7] >;

C3×C8⋊D7 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes D_7
% in TeX

G:=Group("C3xC8:D7");
// GroupNames label

G:=SmallGroup(336,59);
// by ID

G=gap.SmallGroup(336,59);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,313,79,69,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations

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